Math 408 Homework 5

NOTES:
    (1)  Whenever you are asked to test a hypothesis, state the P-value, whether the P-value is for a one-sided or two-sided test if appropriate (that is, if the statistic has a large-sample normal approximation), and whether you accept or reject H_0.

    (2)  If you use MATLAB to do a problem, include (hard copy of) your MATLAB output AND your MATLAB program in an APPENDIX to your homework. That is, do not mix together the answers to the questions and your computer output. In that way, for problems in which you used MATLAB, your answers become an ``executive summary'' that gives your conclusions, and interested parties can then look or not look at your actual MATLAB code and output to get more information or to see what happened if you get a wrong answer.

(3) In the following, ^ means superscript, _ (underscore) means subscript, and Sum(i=1,9) means the sum for i=1 to 9.

1.  Fit a loess curve to the environ data using lambda=1,2 and various values for alpha. Compare the curves. What values of the parameters seem to be the best? In making your comparison, look at residual plots and smoothed scatterplots. One thing to look for is excessive structure (wiggliness) in the loess curve that is not supported by the data.

2. Repeat the smooth of the vineyard data using the running mean smoother with N=5,10,15,20. Discuss what happens with the smooth.

3.  Smooth the vineyard data using loess and the kernel estimator. Compare with the smooths obtained using the other methods.

4. Cruess (1989) points out that the error of applying the ordinary chi-squared statistics to paired data occurs frequently in the medical literature. Cruess cites in particular the study of Shen et al. (1988). They compared the results of two tests, ABC-ELISA and standard ELISA, on 101 hydatidosis patients. (Hydatidosis, or hydatid disease, is infestation with echinococcus, a genus of tapeworms.) Shen et al. used the ordinary unpaired chi-squared test and reported a pvalue<0.005. This was inappropriate because each case was tested using both laboratory procedures and thus the data were paired. Instead of the ordinary unpaired chi-square test, McNemar’s test should have been performed. Information on the 101 pairs is given the following table.

Standard ELISA		+	-
ABC-ELISA	+	82	13
	-	6	0

Perform McNemar’s test. What is the p-value? What do you conclude concerning the hypothesis of equal proportions positive for both ELISA tests?

5. For the Diehr et al (1989) study, the liver scan data for hospital 16 are given in the following table. Let p1 be the probability that a black patient in hospital 16 with local and regional disease will receive a liver scan and let p2 be the probability that a white patient in hospital 16 with local or regional disease will receive a liver scan. Test H0: p1=p2 versus the alternative p1>p2 using Fisher’s exact test. What is the pvalue?

	Liver Scan
Patients	Yes	No
Black	2	3
White	3	12